Frobenius theorem (real division algebras)

:Note that if *z* ∈ **C** ∖ **R** then *Q*(*z*; *x*) is irreducible over **R**. ### The claim The key to the argument is the following :**Claim.** The set V of all elements a of D such that *a*2 ≤ 0 is a vector subspace of D of dimension *n* − 1. Moreover as **R**-vector spaces, which implies that V generates D as an algebra. **Proof of Claim:** Pick a in D with characteristic polynomial *p*(*x*). By the fundamental theorem of algebra, we can write :p(x) = (x-t_1)\cdots(x-t_r) (x-z_1)(x - \overline{z_1}) \cdots (x-z_s)(x - \overline{z_s}), \qquad t_i \in \mathbf{R}, \quad z_j \in \mathbf{C} \setminus \mathbf{R}. We can rewrite *p*(*x*) in terms of the polynomials *Q*(*z*; *x*): :p(x) = (x-t_1)\cdots(x-t_r) Q(z_1; x) \cdots Q(z_s; x). Since *zj* ∈ **C** ∖ **R**, the polynomials *Q*(*zj*; *x*) are all irreducible over **R**. By the Cayley–Hamilton theorem, and because D is a division algebra, it follows that either for some i or that for some j. The first case implies that a is real. In the second case, it follows that *Q*(*zj*; *x*) is the minimal polynomial of a. Because *p*(*x*) has the same complex roots as the minimal polynomial and because it is real it follows that :p(x) = Q(z_j; x)^k = \left(x^2 - 2\operatorname{Re}(z_j) x + |z_j|^2 \right)^k for some *k*. Since *p*(*x*) is the characteristic polynomial of a the coefficient of *x* 2*k* − 1 in *p*(*x*) is tr(*a*) up to a sign. Therefore, we read from the above equation we have: if and only if , in other words if and only if {{math|*a*2 −*zj*2 So V is the subset of all a with . In particular, it is a vector subspace. The rank–nullity theorem then implies that V has dimension *n* − 1 since it is the kernel of \operatorname{tr} : D \to \mathbf{R}. Since **R** and V are disjoint (i.e. they satisfy \mathbf R \cap V = \{0\}), and their dimensions sum to n, we have that . ### The finish For *a*, *b* in V define . Because of the identity , it follows that *B*(*a*, *b*) is real. Furthermore, since *a*2 ≤ 0, we have: *B*(*a*, *a*) 0 for *a* ≠ 0. Thus B is a positive-definite symmetric bilinear form, in other words, an inner product on V. Let W be a subspace of V that generates D as an algebra and which is minimal with respect to this property. Let *e*1, ..., *ek* be an orthonormal basis of W with respect to *B*. Then orthonormality implies that: :e_i^2 =-1, \quad e_i e_j = - e_j e_i. The form of D then depends on k: If , then D is isomorphic to **R**. If , then D is generated by 1 and *e*1 subject to the relation . Hence it is isomorphic to **C**. If , it has been shown above that D is generated by 1, *e*1, *e*2 subject to the relations :e_1^2 = e_2^2 =-1, \quad e_1 e_2 = - e_2 e_1, \quad (e_1 e_2)(e_1 e_2) =-1. These are precisely the relations for **H**. If *k* 2, then D cannot be a division algebra. Assume that *k* 2. Define and consider . By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that . If D were a division algebra, implies , which in turn means: and so *e*1, ..., *e**k*−1 generate D. This contradicts the minimality of W. ## Remarks and related results - The fact that D is generated by *e*1, ..., *ek* subject to the above relations means that D is the Clifford algebra of **R***n*. The last step shows that the only real Clifford algebras which are division algebras are Cℓ0, Cℓ1 and Cℓ2. - As a consequence, the only commutative division algebras are **R** and **C**. Also note that **H** is not a **C**-algebra. If it were, then the center of **H** has to contain **C**, but the center of **H** is **R**. - - This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are **R**, **C**, **H**, and the (non-associative) algebra **O**. - **Pontryagin variant.** If D is a connected, locally compact division ring, then , or **H**. ## References - Ray E. Artz (2009) [Scalar Algebras and Quaternions](http://www.math.cmu.edu/~wn0g/noll/qu1.pdf), Theorem 7.1 "Frobenius Classification", page 26. - Ferdinand Georg Frobenius (1878) "[Über lineare Substitutionen und bilineare Formen](http://commons.wikimedia.org/wiki/File:%C3%9Cber_lineare_Substitutionen_und_bilineare_Formen.djvu)", *Journal für die reine und angewandte Mathematik* 84:1–63 (Crelle's Journal). Reprinted in *Gesammelte Abhandlungen* Band I, pp. 343–405. - Yuri Bahturin (1993) *Basic Structures of Modern Algebra*, Kluwer Acad. Pub. pp. 30–2 . - Leonard Dickson (1914) *Linear Algebras*, Cambridge University Press. See §11 "Algebra of real quaternions; its unique place among algebras", pages 10 to 12. - R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366–8. - Lev Semenovich Pontryagin, Topological Groups, page 159, 1966. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)?action=history). ::